Finally, in this section we calculate the gate errors of the time-optimal pulses found above at a specific blockade strength and decay rate of the Rydberg state. The same setup as in [1]} is chosen, with the qubits encoded in the clock states of Caesium and the Rydberg state \(\mathinner {|{r}\rangle } = \mathinner {|{107P_{3/2}\, m_j=3/2}\rangle }\) . At 300K the Rydberg state has a lifetime of \(1/\Gamma = 540\mu \mathrm {s}\) . Adding the error contributions from the decay of the Rydberg state and from blockade leakage and neglecting all other error sources, the gate error is given by
\(1-F = \frac{\Gamma (T_R\Omega _{\mathrm {max}})}{\Omega _{\mathrm {max}}} + \frac{\Omega _{\mathrm {max}}^2 \alpha }{B^2(T\Omega _{\mathrm {max}})^2}.\) 
